Here's a comparison between 965 BE and a 2500k. None of them overclocked.
http://www.anandtech.com/bench/Product/102?vs=288
An overclocked (4.0-4.1 Ghz) 965 BE is still stable and comes close to a stock 2500k. Close means not it's as fast, just ... close. And that's still a stock 2500k with a lot of headroom when overclocked itself. Not to mention the power draw is quite a bit higher on a Phenom (125W stock, even more OC'ed).
That being said, you can get a mobo (970 chipset) + 965BE + 8GB (1866) RAM for the price of just the 2500k.
None of the Bulldozer chips is really an option, because the fastest one is still short in performance of a 2500k and just a little bit cheaper. Not as much cheaper as a 965, so price advantage goes out of the window compared to the performance gain you would get by spending a few more bucks on that 2500k.
Note: I'm comparing to a 2500k because it is a few bucks cheaper than an 3570k and almost the same in performance, with a slightly higher power draw. If you can get the 3570k for about the same price, take it over the 2500k.
Considering your budget, a 965 BE leaves more room for a nice overall system. Like an SSD + good case or a 7870 or a quality modular PSU for a clean and stable system ...
For gaming on 1080p with a 7850/70 a Phenom is fine, few fps slower, but fine.